Optimal. Leaf size=68 \[ \frac{A}{3 a^2 \left (a+b x^3\right )}-\frac{A \log \left (a+b x^3\right )}{3 a^3}+\frac{A \log (x)}{a^3}+\frac{A b-a B}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.0583117, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ \frac{A}{3 a^2 \left (a+b x^3\right )}-\frac{A \log \left (a+b x^3\right )}{3 a^3}+\frac{A \log (x)}{a^3}+\frac{A b-a B}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x \left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A}{a^3 x}+\frac{-A b+a B}{a (a+b x)^3}-\frac{A b}{a^2 (a+b x)^2}-\frac{A b}{a^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{A b-a B}{6 a b \left (a+b x^3\right )^2}+\frac{A}{3 a^2 \left (a+b x^3\right )}+\frac{A \log (x)}{a^3}-\frac{A \log \left (a+b x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.0450223, size = 59, normalized size = 0.87 \[ \frac{\frac{a \left (-a^2 B+3 a A b+2 A b^2 x^3\right )}{b \left (a+b x^3\right )^2}-2 A \log \left (a+b x^3\right )+6 A \log (x)}{6 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 68, normalized size = 1. \begin{align*}{\frac{A}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{A\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}}+{\frac{A}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{B}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{A\ln \left ( x \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961193, size = 104, normalized size = 1.53 \begin{align*} \frac{2 \, A b^{2} x^{3} - B a^{2} + 3 \, A a b}{6 \,{\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} - \frac{A \log \left (b x^{3} + a\right )}{3 \, a^{3}} + \frac{A \log \left (x^{3}\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49971, size = 250, normalized size = 3.68 \begin{align*} \frac{2 \, A a b^{2} x^{3} - B a^{3} + 3 \, A a^{2} b - 2 \,{\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \log \left (x\right )}{6 \,{\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.10972, size = 75, normalized size = 1.1 \begin{align*} \frac{A \log{\left (x \right )}}{a^{3}} - \frac{A \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3}} + \frac{3 A a b + 2 A b^{2} x^{3} - B a^{2}}{6 a^{4} b + 12 a^{3} b^{2} x^{3} + 6 a^{2} b^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10443, size = 100, normalized size = 1.47 \begin{align*} -\frac{A \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{A \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{3 \, A b^{3} x^{6} + 8 \, A a b^{2} x^{3} - B a^{3} + 6 \, A a^{2} b}{6 \,{\left (b x^{3} + a\right )}^{2} a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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